Questions - OCR - A-Level Maths

OCR Further Pure Core 1 Specimen Q9

5 marks Standard +0.8

9 Prove by induction that, for all positive integers $n$, $$\sum _ { r = 1 } ^ { n } \frac { 5 - 4 r } { 5 ^ { r } } = \frac { n } { 5 ^ { n } }$$

OCR Further Pure Core 1 Specimen Q10

10 marks Standard +0.3

10 The Argand diagram below shows the origin $O$ and pentagon $A B C D E$, where $A , B , C , D$ and $E$ are the points that represent the complex numbers $a , b , c , d$ and $e$, and where $a$ is a positive real number. You are given that these five complex numbers are the roots of the equation $z ^ { 5 } - a ^ { 5 } = 0$. \includegraphics[max width=\textwidth, alt={}, center]{94ecfc6e-df52-45a0-8f7b-f33fda391b15-4_903_883_477_502}

Justify each of the following statements.
(a) $A , B , C , D$ and $E$ lie on a circle with centre $O$.
(b) $A B C D E$ is a regular pentagon.
(c) $b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c$ (d) $b ^ { * } = e$ (e) $a + b + c + d + e = 0$
The midpoints of sides $A B , B C , C D , D E$ and $E A$ represent the complex numbers $p , q , r , s$ and $t$. Determine a polynomial equation, with real coefficients, that has roots $p , q , r , s$ and $t$.

OCR Further Pure Core 1 Specimen Q11

19 marks Challenging +1.2

11 A company is required to weigh any goods before exporting them overseas. When a crate is placed on a set of weighing scales, the mass displayed takes time to settle down to its final value. The company wishes to model the mass, $m \mathrm {~kg}$, which is displayed $t$ seconds after a crate X is placed on the scales.
For the displayed mass it is assumed that the rate of change of the quantity $\left( 0.5 \frac { \mathrm {~d} m } { \mathrm {~d} t } + m \right)$ with respect to time is proportional to $( 80 - m )$.

Show that $\frac { \mathrm { d } ^ { 2 } m } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} m } { \mathrm {~d} t } + 2 \mathrm {~km} = 160 \mathrm { k }$, where $k$ is a real constant. It is given that the complementary function for the differential equation in part (i) is $\mathrm { e } ^ { \lambda t } ( A \cos 2 t + B \sin 2 t )$, where $A$ and $B$ are arbitrary constants.
Show that $k = \frac { 5 } { 2 }$ and state the value of the constant $\lambda$. When X is initially placed on the scales the displayed mass is zero and the rate of increase of the displayed mass is $160 \mathrm {~kg} \mathrm {~s} ^ { - 1 }$.
Find $m$ in terms of $t$.
Describe the long term behaviour of $m$.
With reference to your answer to part (iv), comment on a limitation of the model.
(a) Find the value of $m$ that corresponds to the stationary point on the curve $m = \mathrm { f } ( t )$ with the smallest positive value of $t$.
(b) Interpret this value of $m$ in the context of the model.
Adapt the differential equation $\frac { \mathrm { d } ^ { 2 } m } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} m } { \mathrm {~d} t } + 5 m = 400$ to model the mass displayed $t$ seconds after a crate Y , of mass 100 kg , is placed on the scales. \section*{END OF QUESTION PAPER} \section*{Copyright Information:} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

OCR Further Pure Core 2 2019 June Q1

7 marks Standard +0.3

1 In this question you must show detailed reasoning.

By using partial fractions show that $\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } + 3 r + 2 } = \frac { 1 } { 2 } - \frac { 1 } { n + 2 }$.
Hence determine the value of $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } + 3 r + 2 }$.

OCR Further Pure Core 2 2019 June Q2

8 marks Standard +0.3

2

A plane $\Pi$ has the equation $\mathbf { r } \cdot \left( \begin{array} { r } 3 \\ 6 \\ - 2 \end{array} \right) = 15 . C$ is the point $( 4 , - 5,1 )$.
Find the shortest distance between $\Pi$ and $C$.
Lines $l _ { 1 }$ and $l _ { 2 }$ have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 4

OCR Further Pure Core 2 2019 June Q4

5 marks Standard +0.3

4
3
1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
4
- 2 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }

OCR Further Pure Core 2 2019 June Q5

11 marks

5
2
4 \end{array} \right) + \mu \left( \begin{array} { r } 1
- 2
1 \end{array} \right) \end{aligned}$$ Find, in exact form, the distance between $l _ { 1 }$ and $l _ { 2 }$.

OCR Further Pure Core 2 2019 June Q7

7 marks Standard +0.8

7 In an Argand diagram the points representing the numbers $2 + 3 \mathrm { i }$ and $1 - \mathrm { i }$ are two adjacent vertices of a square, $S$.

Find the area of $S$.
Find all the possible pairs of numbers represented by the other two vertices of $S$.

OCR Further Pure Core 2 2019 June Q8

8 marks Challenging +1.2

8 In this question you must show detailed reasoning.

By writing $\sin \theta$ in terms of $\mathrm { e } ^ { \mathrm { i } \theta }$ and $\mathrm { e } ^ { - \mathrm { i } \theta }$ show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta ) .$$
Hence show that $\sin \frac { 1 } { 8 } \pi = \frac { 1 } { 2 } \sqrt [ 6 ] { 20 - 14 \sqrt { 2 } }$.

OCR Further Pure Core 2 2019 June Q10

7 marks Challenging +1.3

10

Use differentiation to find the first two non-zero terms of the Maclaurin expansion of $\ln \left( \frac { 1 } { 2 } + \cos x \right)$.
By considering the root of the equation $\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0$ deduce that $\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }$. \section*{END OF QUESTION PAPER}

OCR Further Pure Core 2 2022 June Q1

6 marks Moderate -0.3

1

Find a vector which is perpendicular to both $3 \mathbf { i } - 5 \mathbf { j } - \mathbf { k }$ and $\mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }$. The equations of two lines are $\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$ and $\mathbf { r } = \mathbf { i } + 11 \mathbf { j } - 4 \mathbf { k } + \mu ( - \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )$.
Show that the lines intersect, stating the point of intersection.

OCR Further Pure Core 2 2022 June Q2

5 marks Standard +0.3

2 Two polar curves, $C _ { 1 }$ and $C _ { 2 }$, are defined by $C _ { 1 } : r = 2 \theta$ and $C _ { 2 } : r = \theta + 1$ where $0 \leqslant \theta \leqslant 2 \pi$. $C _ { 1 }$ intersects the initial line at two points, the pole and the point $A$.

Write down the polar coordinates of $A$.
Determine the polar coordinates of the point of intersection of $C _ { 1 }$ and $C _ { 2 }$. The diagram below shows a sketch of $C _ { 1 }$. \includegraphics[max width=\textwidth, alt={}, center]{007f07ee-cb29-4a97-93d9-2328079c4aea-2_681_1353_1318_244}
On the copy of this sketch in the Printed Answer Booklet, sketch $C _ { 2 }$.

OCR Further Pure Core 2 2022 June Q4

4 marks Standard +0.8

4 In this question you must show detailed reasoning.
Determine the smallest value of $n$ for which $\frac { 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } } { 1 + 2 + \ldots + n } > 341$.

OCR Further Pure Core 2 2022 June Q5

7 marks Standard +0.3

5

By using the exponential definitions of $\sinh x$ and $\cosh x$, prove the identity $\cosh 2 x \equiv \cosh ^ { 2 } x + \sinh ^ { 2 } x$.
Hence find an expression for $\cosh 2 x$ in terms of $\cosh x$.
Determine the solutions of the equation $5 \cosh 2 x = 16 \cosh x + 21$, giving your answers in exact logarithmic form.

OCR Further Pure Core 2 2022 June Q6

10 marks Challenging +1.2

6 A particle, $P$, positioned at the origin, $O$, is projected with a certain velocity along the $x$-axis. $P$ is then acted on by a single force which varies in such a way that $P$ moves backwards and forwards along the $x$-axis. When the time after projection is $t$ seconds, the displacement of $P$ from the origin is $x \mathrm {~m}$ and its velocity is $v \mathrm {~ms} ^ { - 1 }$. The motion of $P$ is modelled using the differential equation $\ddot { x } + \omega ^ { 2 } x = 0$, where $\omega$ rads $^ { - 1 }$ is a positive constant.

Write down the general solution of this differential equation. $D$ is the point where $x = d$ for some positive constant, $d$. When $P$ reaches $D$ it comes to instantaneous rest.
Using the answer to part (a), determine expressions, in terms of $\omega$, $d$ and $t$ only, for the following quantities
- $X$
- $v$
- Hence show that, according to the model, $v ^ { 2 } = \omega ^ { 2 } \left( d ^ { 2 } - x ^ { 2 } \right)$.
The quantity $z$ is defined by $z = \frac { 1 } { v }$.
Using part (c), determine an expression for $\mathrm { Z } _ { \mathrm { m } }$, the mean value of z with respect to the displacement, as $P$ moves directly from $O$ to $D$. One measure of the validity of the model is consideration of the value of $\mathrm { z } _ { \mathrm { m } }$. If $\mathrm { z } _ { \mathrm { m } }$ exceeds 8 then the model is considered to be valid. The value of $d$ is measured as 0.25 to 2 significant figures. The value of $\omega$ is measured as $0.75 \pm 0.02$.
Determine what can be inferred about the validity of the model from the given information.
Find, according to the model, the least possible value of the velocity with which $P$ was initially projected. Give your answer to $\mathbf { 2 }$ significant figures.

OCR Further Pure Core 2 2022 June Q7

13 marks Standard +0.8

7 You are given that $a$ is a parameter which can take only real values.
The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { c c r } 2 & 4 & - 6 \\ - 3 & 10 - 4 a & 9 \\ 7 & 4 & 4 \end{array} \right)$.

Find an expression for the determinant of $\mathbf { A }$ in terms of $a$. You are given the following system of equations in $x , y$ and $z$. $$\begin{array} { r r } 2 x + & 4 y - 6 z = \\ - 3 x + & ( 10 - 4 a ) y + 9 z = \\ 7 x + & 4 y + 4 z = \\ 7 x + & 11 \end{array}$$ The system can be written in the form $\mathbf { A } \left( \begin{array} { c } \mathrm { x } \\ \mathrm { y } \\ \mathrm { z } \end{array} \right) = \left( \begin{array} { r } 6 \\ - 9 \\ 11 \end{array} \right)$.
1. In the case where $\mathbf { A }$ is not singular, solve the given system of equations by using $\mathbf { A } ^ { - 1 }$.
2. In the case where $\mathbf { A }$ is singular describe the configuration of the planes whose equations are the three equations of the system. The transformation represented by $\mathbf { A }$ is denoted by T .
  A 3-D object of volume $| 5 a - 20 |$ is transformed by T to a 3-D image.
1. Determine the range of values of $a$ for which the orientation of the image is the reverse of the orientation of the object.
2. Determine the range of values of $a$ for which the volume of the image is less than the volume of the object.

OCR Further Pure Core 2 2022 June Q10

8 marks Challenging +1.8

10 The coordinates of the points $A$ and $B$ are ( $3 , - 2 , - 1$ ) and ( $13,10,9$ ) respectively.

The plane $\Pi _ { A }$ contains $A$ and the plane $\Pi _ { B }$ contains $B$.
The planes $\Pi _ { A }$ and $\Pi _ { B }$ are parallel.
The $x$ and $y$ components of any normal to plane $\Pi _ { A }$ are equal.
The shortest distance between $\Pi _ { A }$ and $\Pi _ { B }$ is 2 .

There are two possible solution planes for $\Pi _ { A }$ which satisfy the above conditions.
Determine the acute angle between these two possible solution planes.

OCR Further Pure Core 2 2023 June Q1

8 marks Easy -1.3

1

The matrix $\mathbf { P }$ is given by $\mathbf { P } = \left( \begin{array} { l l l l } 1 & 0 & - 2 & 2 \\ 4 & 2 & - 2 & 3 \end{array} \right)$.
1. Write down the dimensions of $\mathbf { P }$.
2. Write down the transpose of $\mathbf { P }$.
The matrices $\mathbf { Q } , \mathbf { R }$ and $\mathbf { S }$ are given by $\mathbf { Q } = \left( \begin{array} { l l } 1 & 2 \end{array} \right) , \mathbf { R } = \left( \begin{array} { r r } 3 & - 4 \\ 2 & 3 \end{array} \right)$ and $\mathbf { S } = \left( \begin{array} { l l } 3 & - 2 \end{array} \right)$. Write down the sum of the two of these matrices which are conformable for addition.
The dimensions of matrix $\mathbf { A }$ are 4 by 5. The matrices $\mathbf { A }$ and $\mathbf { B }$ are conformable for multiplication so that the matrix $\mathbf { C } = \mathbf { B A }$ can be formed. The matrix $\mathbf { C }$ has 6 rows.
1. Write down the number of columns that $\mathbf { C }$ has.
2. Write down the dimensions of $\mathbf { B }$.
3. Explain whether the matrix $\mathbf { A B }$ can be formed.
Find the value of $c$ for which $\left( \begin{array} { r r } - 2 & 3 \\ 6 & 10 \end{array} \right) \left( \begin{array} { r r } c & 5 \\ 10 & 13 \end{array} \right) = \left( \begin{array} { r r } c & 5 \\ 10 & 13 \end{array} \right) \left( \begin{array} { r r } - 2 & 3 \\ 6 & 10 \end{array} \right)$.

OCR Further Pure Core 2 2023 June Q2

7 marks Standard +0.3

2 In this question you must show detailed reasoning.

Write the complex number $- 24 + 7 \mathrm { i }$ in modulus-argument form.
Solve the simultaneous equations given below, giving your answers in cartesian form. $$\begin{aligned} i z + 3 w & = - 7 i \\ - 6 z + 5 i w & = 3 + 13 i \end{aligned}$$

OCR Further Pure Core 2 2023 June Q3

6 marks Standard +0.3

3

Show that $\frac { \mathrm { d } } { \mathrm { d } u } \left( \sinh ^ { - 1 } u \right) = \frac { 1 } { \sqrt { u ^ { 2 } + 1 } }$.
Find the equation of the normal to the graph of $\mathrm { y } = \sinh ^ { - 1 } 2 \mathrm { x }$ at the point where $x = \sqrt { 6 }$. Give your answer in the form $\mathrm { y } = \mathrm { mx } + \mathrm { c }$ where $m$ and $c$ are given in exact, non-hyperbolic form.

OCR Further Pure Core 2 2023 June Q6

8 marks Standard +0.8

6 The equation of the plane $\Pi$ is $\mathbf { r } = \left( \begin{array} { r } - 1 \\ 2 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { l } 4 \\ 4 \\ 3 \end{array} \right) + \mu \left( \begin{array} { r } - 2 \\ 3 \\ 1 \end{array} \right)$.

Find the acute angle between $\Pi$ and the plane with equation $\mathbf { r } . \left( \begin{array} { l } 2 \\ 0 \\ 3 \end{array} \right) = 4$. The point $A$ has coordinates ( $9 , - 7,20$ ).
The point $F$ is the point of intersection between $\Pi$ and the perpendicular from $A$ to $\Pi$.
Determine the coordinates of $F$.

OCR Further Pure Core 2 2023 June Q7

8 marks Challenging +1.8

7 In this question you must show detailed reasoning.

Show that $$\sum _ { r = 1 } ^ { n } \frac { 5 r + 6 } { r ^ { 3 } + r ^ { 2 } } = \frac { a } { n + 1 } + b + c \sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } }$$ where $a$, $b$ and $c$ are integers whose values are to be determined. You are given that $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }$ exists and is equal to $\frac { 1 } { 6 } \pi ^ { 2 }$.
Show that $\sum _ { r = 1 } ^ { \infty } \frac { 5 r + 6 } { r ^ { 3 } + r ^ { 2 } }$ exists and is equal to $( \pi - 1 ) ( \pi + 1 )$.

OCR Further Pure Core 2 2023 June Q8

11 marks

8 A surge in the current, $I$ units, through an electrical component at a time, $t$ seconds, is to be modelled. The surge starts when $t = 0$ and there is initially no current through the component. When the current has surged for 1 second it is measured as being 5 units. While the surge is occurring, $I$ is modelled by the following differential equation. $\left( 2 t - t ^ { 2 } \right) \frac { d l } { d t } = \left( 2 t - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } - 2 ( t - 1 ) l$

By using an integrating factor show that, according to the model, while the surge is occurring, $I$ is given by $\mathrm { I } = \left( 2 \mathrm { t } - \mathrm { t } ^ { 2 } \right) \left( \sin ^ { - 1 } ( \mathrm { t } - 1 ) + 5 \right)$. The surge lasts until there is again no current through the component.
Determine the length of time that the surge lasts according to the model.
Determine, according to the model, the rate of increase of the current at the start of the surge. Give your answer in an exact form.

OCR Further Pure Core 2 2023 June Q9

9 marks Challenging +1.2

9 A function is defined by $y = f ( t )$ where $f ( t ) = \ln ( 1 + a t )$ and $a$ is a constant.

By considering $\frac { d y } { d t } , \frac { d ^ { 2 } y } { d t ^ { 2 } } , \frac { d ^ { 3 } y } { d t ^ { 3 } }$ and $\frac { d ^ { 4 } y } { d t ^ { 4 } }$, make a conjecture for a general formula for $\frac { d ^ { n } y } { d t ^ { n } }$ in terms of $n$ and $a$ for any integer $n \geqslant 1$.
Use induction to prove the formula conjectured in part (a).
In the case where $\mathrm { f } ( t ) = \ln ( 1 + 2 t )$, find the rate at which the $6 ^ { \text {th } }$ derivative of $\mathrm { f } ( t )$ is varying when $t = \frac { 3 } { 2 }$.

OCR Further Pure Core 2 2024 June Q1

5 marks Moderate -0.5

1

Use the method of differences to show that $\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \left( \frac { 1 } { \mathrm { r } } - \frac { 1 } { \mathrm { r } + 1 } \right) = 1 - \frac { 1 } { \mathrm { n } + 1 }$.
Hence determine the following sums.
1. $\quad \sum _ { r = 1 } ^ { 99 } \frac { 1 } { r } - \frac { 1 } { r + 1 }$
2. $\quad \sum _ { r = 100 } ^ { \infty } \frac { 1 } { r } - \frac { 1 } { r + 1 }$

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OCR Further Pure Core 1 Specimen Q9

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